In channels with steep slopes, the flow is supercritical and the formation of roll waves is expected. In order to evaluate the flow behaviors in steep open channels, the Azad dam spillway is simulated numerically and its flow characteristics are compared with the physical model tests. In this study, the hydraulic variables including hydraulic depth and water surface profile are studied along the steep open channels. The k-ɛ turbulence model is considered in this study. It is shown that in the conditions in which the angle of the wave forehead is less than 35 degrees, the formation of roll waves is inevitable.

INTRODUCTION

Usually the regime of flow in the steep open channels is supercritical. In supercritical flows, the instability on the free surface is intensified, which is the prerequisite for roll wave formation. Roll waves in an open channel are formed in which the gravity forces are dominant. This phenomenon might be studied by numerical and physical models. Nowadays, numerical models are used more frequently than experimental models as they allow the spending of less time and money.

Hager (1992) studied classic hydraulic jump. The effect of roughness on the hydraulic jump length is studied. His results show that the roughness has a remarkable influence on jump characteristics. Interaction of the waves and turbulence generated by wind has been extensively investigated, both in physical models (Zhang 1995) and in numerical studies (Borue et al. 1995). In the studies by Gualtieri & Chanson (2007) the distributions of the void fraction and the air bubbles count rate were recorded in the hydraulic jump for the Froude numbers varying from 5.2 to 14.3, which is consistent with previous studies. Also, the data demonstrated that the upper boundary of the air diffusion layer tends systematically to linearly increase by increasing the distance from the jump toe for a fixed inflow Froude number expressed as follows: 
formula
1
In the above equation, Fr represents Froude number, g gravitational acceleration, V and y are the velocity and depth of flow, respectively.

A comprehensive study for simulating flow over a stepped spillway, especially upstream of the inception point of air entrainment was achieved by Bombardelli et al. (2011) and Chakib (2013), using FLUENT software and a volume of fluid model. They found a good agreement between the experimental data and the numerical results for velocity vectors, streamlines and total pressures over the spillway. Bazargan & Aghebatie (2015) used the standard k-ɛ model to simulate turbulence and volume of fluid model to track the free surface of the chute for different discharges in a three-dimensional numerical model. Velocities and water surface profiles obtained from a numerical model are compared with the physical model data. The results showed that the numerical results and the measured data are basically consistent and can provide a reliable basis for chute optimization design. Also, they showed that for rectangular chutes with the ratio of hydraulic depth over the chute width interior of 0.16, the flow becomes unstable and roll waves are formed. Bayon-Barrachina & López-Jiménez (2015) studied the hydraulic jump in a horizontal rectangular canal by using OpenFOAM software and different turbulence models such as standard k − ɛ, RNG k − ɛ, and SST k −ω turbulence models. Accuracies of about 98% are achieved in the numerical model for different parameters such as sequent depths, efficiency, roller length, free surface profile, etc. in comparison with the physical model tests. In the present study, Azad Dam spillway is simulated numerically and compared with the prototype results. Design essentials and criteria for producing the roll waves are investigated. In this study, the previous criteria are evaluated and further criteria are proposed using the FLUENT software.

METHODS

Data provided for numerical simulation

The data provided from the physical model tests of the Azad Dam spillway built in the Hydraulic Structures Department of Iran Water Research Institute with the scale of 1:33.33, are converted to prototype and compared with the numerical results. The corresponding discharges tested in the hydraulic laboratory were 500 and 800 m3/s. The spillway consisted of an ogee of 9.78 m length at 1,465 m above sea level. The ogee curve is defined by , so the beginning level of the chute was located 1,458.13 m above sea level. The chute is 30 m in width and has two different slopes; 5% and 36.4% with horizontal lengths equal to 45.4 and 198.31 m in each part, respectively. The chute ends in a flip bucket with a radius of 15 m and a horizontal length of 10.58 m. A schematic view of the spillway is shown in Figure 1.
Figure 1

Longitudinal section of Azad Dam spillway.

Figure 1

Longitudinal section of Azad Dam spillway.

Simulation of free surface flow

The applied numerical model is one of the most complete computational fluid dynamic models in which the equations are solved using the finite volume method. In order to simulate the water surface of the flow, the volume of fluid is used. This formulation relies on the principle that two or more phases are not interpenetrating. In each control volume, the sum of the volume fractions for all phases is unity. The fields for all variables and properties are shared by the phases and represent volume-averaged values, as long as the volume fraction of each of the phases is known at each location. Thus, the variables and properties in any given cell are either purely representative of one of the phases, or representative of a mixture of the phases, depending upon the volume fraction values. In the volume of fluid (VOF) model, the momentum equation is solved in the whole domain and then divided between the different phases. Due to the existence of and μ parameters, the momentum equation is a function of the volumetric ratios of all phases (FLUENT 2004; Houichi et al. 2006; Yeoh & Tu 2010). 
formula
2
in which, is the fluid density, μ,, and are the dynamic viscosity, fluid pressure, velocity vector and finally the gravitational body force, respectively. Also, in the above equations, is the transpose sign.
The velocity profile of the turbulent flow is highly influenced by the wall effect. Therefore an accurate description of the velocity distribution near the wall is of particular importance. The wall y+ and U* are non-dimensional numbers presented in Equations (3) and (4), which determine whether the influences in the wall adjacent cells are laminar or turbulent, hence indicating the part of the turbulent boundary layer that they resolve (Akoz & Kirkgoz 2009). 
formula
3
 
formula
4
In the above equations, is the average velocity, y is the distance from the wall and is the wall stress. The logarithmic velocity law for is valid, which in FLUENT software separates this boundary, and the following equation governs this layer. 
formula
5
In the above equation k and E are the Von Karman and wall function constants and their values are 0.42 and 9.81, respectively. For , the governing linear stress-strain equation is as follows: 
formula
6

Turbulence model

One of the main characteristics of turbulent flow is the fluctuating velocity fields. These fluctuations cause mixing of transported quantities like momentum, energy and species concentration and thereby also fluctuations in the transported quantities. Because of the small scales and high frequencies of the fluctuations, they are too computationally expensive to simulate directly in practical engineering situations. Instead, the instantaneous governing equations are time-averaged to remove the small scales and the result is a set of less expensive equations containing additional unknown variables. These unknown (turbulence) variables are determined in terms of modeled variables in turbulence models.

The turbulence model that is used in the present study is the standard k-ɛ model. It is one of the two-equation models which are considered the simplest of the so called complete models of turbulence. Ever since it was proposed in 1974, its popularity in industrial flow simulations has been explained by its robustness, economy and reasonable accuracy for a wide range of turbulent flows. The model is a semi-empirical model based on modeled transport equations for the turbulence kinetic energy (k) and its dissipation rate (ɛ). Applied equations in the model are as follows (Kherbache et al. 2013; Kositgittiwong et al. 2013): 
formula
7
 
formula
8
in which μ indicates the dynamic viscosity, is the fluid density, is the eddy viscosity, is an empirical coefficient equal to 0.09. The coefficients of the above equations are . is a constant considered 1 for the flow component parallel to gravity direction and equals zero for the flow components perpendicular to the gravity direction. Terms and are the turbulent Prandtl numbers for and , respectively. 
formula
9
 
formula
10

In the above equations, b, T and are the coefficient of thermal expansion, the temperature and the eddy viscosity, respectively. Term is the turbulent Prandtl number for energy and the variables and are velocities in the and directions, finally and correspond to x and y coordinates.

Verification of numerical model

Selection of boundary conditions, turbulence model and the cell sizes introduces errors in the numerical models. In order to achieve the optimum composition of all conditions, the numerical results must be compared with the physical model tests. In this research, both the velocity and hydraulic depth obtained from the numerical model are compared with their prototype values produced from the hydraulic model. Figures 2 and 3 show the comparisons of the velocity and depth changes, respectively, with the discharges of 500 and 800 m3/s over the spillway, chute and flip bucket of Azad Dam based on numerical modeling and experimental simulation results. As shown in these figures, a good agreement is achieved between the numerical and the hydraulic model tests. The mean relative errors of water surface profiles between simulated and experimental values for standard k-ɛ, RNG k-ɛ, and realizable k-ɛ turbulence models are about 6.2%–9.33% and 11.4%, respectively. The turbulence model that is used in the present study is the standard k-ɛ model.
Figure 2

Comparison of the velocity distribution curves between the experimental and numerical models of the Azad Dam spillway.

Figure 2

Comparison of the velocity distribution curves between the experimental and numerical models of the Azad Dam spillway.

Figure 3

Comparison of the depth change curves between the experimental and numerical models of the Azad Dam spillway.

Figure 3

Comparison of the depth change curves between the experimental and numerical models of the Azad Dam spillway.

Further simulations

After accomplishing the verification tests, the study is focused on the different slopes and shapes. The flow field was modeled in Gambit using unequal hexahedron rectangular elements. Then, the coordinates of the created nodes (not the created rectangular mesh grid) were exported from Gambit into FLUENT. Steep open channels modeled in this study are considered as three-dimensional channels that discharge 15 to 30 m3/s and the longitudinal slope varying from 17 to 20%. Inlet boundary condition was defined as mass-flow-inlet and as zero pressure for the outlet and upper space of the models. The number of elements varied from about 172,238 for the channel with the diameter of 4.5 m and a slope of 17% and a flow discharge of 30 m3/s, up to about 137,745 for the channel with the diameter of 3.6 m and a slope of 20% and input discharge of 15 m3/s. The logarithmic distribution of velocity is valid only for the ranging from 30 to 300 according to Kawai & Larsson (2012). The mesh sizes are organized to produce situated in the valuable ranges.

To include the Manning roughness coefficient effect in the model with the scale of 1:33.33, roughness height of 0.085 mm which is equivalent to 0.18 mm (according to Mɛ = ML0.166 in which Mɛ and ML are the roughness scale and length scale, respectively) was considered for the prototype. Time steps varied from 0.0001 up to 0.01 s from the beginning of simulation to the end of the simulation. This value obtained is about 210. A schematic view of the grid mesh with the optimum mesh size is shown in Figure 4.
Figure 4

Schematic view of the grid mesh for the numerical model of the Azad Dam spillway.

Figure 4

Schematic view of the grid mesh for the numerical model of the Azad Dam spillway.

Grid dimensions are considered from 0.07 to 0.15 m in the real model. In the numerical model, the steady state is evaluated by analyzing the vertical distribution of the velocity at the given points in which the input and output discharges must be equivalent. A maximum difference of 0.05% between input and output discharges was considered for the steady state conditions.

RESULTS AND DISCUSSION

The side wall height of the canal is obtained from the longitudinal profile of the water surface, considering the free board. All the depth values in the spillway for 60 points along the models were calculated. Given that the depth values in all points are less than 0.5 m, we conclude that there's a possibility that roll waves will occur at these points, and an instant discharge up to twice the normal discharge which causes the water to overflow from open channel sidewalls. Shown in Figure 5(a) is the roll waves formation for the physical model and also, the water surface profile is shown in Figure 5(b) for a rectangular open channel with a width of 3.4 m and flow discharge of 15 m3/s.
Figure 5

The roll wave formation in the hydraulic model (a) and water surface profile (b) for the numerical model with a flow discharge of 15 m3/s.

Figure 5

The roll wave formation in the hydraulic model (a) and water surface profile (b) for the numerical model with a flow discharge of 15 m3/s.

After the validation and calibration tests of the numerical model, using experimental results including flow depth and velocity and wave formation, the numerical model is used to evaluate the hydraulic properties of the flow and the criteria for the formation of rolling waves in different conditions, which are presented in Table 1.

Table 1

Comparison of the flow depth to channel base ratios for the elliptical open channels with a flow discharge of 15 m3/s

Point length (m)
Elliptical channel diameter (c)Flow depth (D)19202530354045
  0.799 0.748 0.600 0.524 0.476 0.441 0.415 
   0.177 0.166 0.133 0.116 0.106 0.098 0.092 
  0.811 0.758 0.607 0.530 0.478 0.447 0.428 
   0.193 0.181 0.144 0.126 0.113 0.106 0.102 
  0.856 0.802 0.643 0.561 0.511 0.473 0.445 
   0.204 0.191 0.153 0.133 0.121 0.112 0.106 
  0.851 0.796 0.637 0.556 0.502 0.469 0.449 
   0.212 0.199 0.159 0.139 0.125 0.117 0.112 
  0.945 0.885 0.707 0.618 0.558 0.522 0.499 
   0.262 0.245 0.196 0.171 0.155 0.145 0.138 
Point length (m)
Elliptical channel diameter (c)Flow depth (D)19202530354045
  0.799 0.748 0.600 0.524 0.476 0.441 0.415 
   0.177 0.166 0.133 0.116 0.106 0.098 0.092 
  0.811 0.758 0.607 0.530 0.478 0.447 0.428 
   0.193 0.181 0.144 0.126 0.113 0.106 0.102 
  0.856 0.802 0.643 0.561 0.511 0.473 0.445 
   0.204 0.191 0.153 0.133 0.121 0.112 0.106 
  0.851 0.796 0.637 0.556 0.502 0.469 0.449 
   0.212 0.199 0.159 0.139 0.125 0.117 0.112 
  0.945 0.885 0.707 0.618 0.558 0.522 0.499 
   0.262 0.245 0.196 0.171 0.155 0.145 0.138 

In all the modeled elliptical open channels with the hydraulic depth to channel base ratio less than 0.14, roll waves in pulsating flows are expected. 
formula
11

Also, for the rectangular steep channels in which the ratio of flow depth to channel width is less than 0.16, the formation of roll waves is inevitable (Bazargan & Aghebatie 2015).

By reducing the longitudinal slope from 20 to 17% and decreasing the diameter of the elliptical channel from 4.5 to 3.6 m, it was observed that the hydraulic depth along the channel is increased. This condition induces the turbulence on flow later than the previous case, by decreasing the longitudinal slope and decreasing the width in the rectangular steep open channel. Hence, the elliptical channels are more effective, because they induce less turbulence. As the chutes have steep longitudinal slopes, so the flow is always supercritical and this causes instability in the flow surface, which is a prerequisite for the roll wave formation. Therefore, if the criteria indicate that roll waves will occur, we can replace the rectangular steep channels section with an elliptical section.

In the study of all numerical models, the results showed that if the angle of the wave forehead with the level vertical (α) is less than 35°, the formation of roll waves is inevitable. 
formula
12
Since the proposed criteria are dimensionless, they could be used as a prediction of turbulent flow. The results of the comparison between various criteria and numerical models results in the roll wave formation moment showed that for the location of the roll wave formation, Aghebatie and Froude number criteria are consistent with the numerical models results.

CONCLUSIONS

This research in conjunction with the many previous studies shows that for turbulent flow over a steep open channel, numerical models are sufficiently advanced to simulate water surface profile on the open channel. In the present study, applying fluid volume to simulate flow free surface and k-ɛ turbulence model, the proposed criteria that can be used as a design condition for chutes or steep open channels. In all the models with elliptical steep open channels with flow depth to channel base ratios of less than 0.14, the formation of roll waves is inevitable. 
formula
13
The derived results indicate that for the sections of the steep open channels structure if the angle of the wave forehead with the level vertical is less than 35°, we must expect the formation of the rolling waves. 
formula
14

Results indicated that among the shapes of the steep open channels section proposed to reduce the roll wave formation probability, the elliptical channel is more effective. If the criteria indicate that roll waves will occur, the shape of the section should be modified to reduce the probability of waves being generated. The rectangular steep open channels section should be replaced with an elliptical section.

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